![]() If our data does not follow a normal distribution, or if the population standard deviation is unknown (and thus in the formula for Z Z Z we substitute the population standard deviation σ \sigma σ with sample standard deviation), then the test statistics Z Z Z is not necessarily normal. By the way, we have the z-score calculator if you want to focus on this value alone. As Z Z Z is the standardization (z-score) of S n / n S_n/n S n / n, we can conclude that the test statistic Z Z Z follows the standard normal distribution N ( 0, 1 ) \mathrm N(0, 1) N ( 0, 1 ), provided that H 0 \mathrm H_0 H 0 is true. x n follows the normal distribution, with mean n μ 0 n \mu_0 n μ 0 and variance n 2 σ n^2 \sigma n 2 σ. ![]() If H 0 \mathrm H_0 H 0 holds, then the sum S n = x 1 . In what follows, the uppercase Z Z Z stands for the test statistic (treated as a random variable), while the lowercase z z z will denote an actual value of Z Z Z, computed for a given sample drawn from N(μ,σ²). Σ \sigma σ is the population standard deviation. Μ 0 \mu_0 μ 0 is the mean postulated in H 0 \mathrm H_0 H 0 ![]() X ˉ \bar x x ˉ is the sample mean, i.e., x ˉ = ( x 1 .
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